Offline reinforcement learning (RL) aims to find performant policies from logged data without further environment interaction. Model-based algorithms, which learn a model of the environment from the dataset and perform conservative policy optimisation within that model, have emerged as a promising approach to this problem. In this work, we present Robust Adversarial Model-Based Offline RL (RAMBO), a novel approach to model-based offline RL. We formulate the problem as a two-player zero sum game against an adversarial environment model. The model is trained to minimise the value function while still accurately predicting the transitions in the dataset, forcing the policy to act conservatively in areas not covered by the dataset. To approximately solve the two-player game, we alternate between optimising the policy and adversarially optimising the model. The problem formulation that we address is theoretically grounded, resulting in a probably approximately correct (PAC) performance guarantee and a pessimistic value function which lower bounds the value function in the true environment. We evaluate our approach on widely studied offline RL benchmarks, and demonstrate that it outperforms existing state-of-the-art baselines.